Network stress-strain isotherms

Monte Carlo computer simulations were also carried out on filled networks [50,61-63] in an attempt to obtain a better molecular interpretation of how such dispersed fillers reinforce elastomeric materials. The approach taken enabled estimation of the effect of the excluded volume of the filler particles on the network chains and on the elastic properties of the networks. In the first step, distribution functions for the end-to-end vectors of the chains were obtained by applying Monte Carlo methods to rotational isomeric state representations of the chains [64], Conformations of chains that overlapped with any filler particle during the simulation were rejected. The resulting perturbed distributions were then used in the three-chain elasticity model [16] to obtain the desired stress-strain isotherms in elongation. 

The equilibrium stress-strain isotherms in elongation, and the swelling ratios in benzene, were measured at 25°C for these networks. Network chain densities calculated from these measurements exceeded the values predicted from stoichiometry. 

The two network precursors and solvent (if present) were combined with 20 ppm catalyst and reacted under argon at 75°C to produce the desired networks. The sol fractions, ws, and equilibrium swelling ratio In benzene, V2m, of these networks were determined according to established procedures ( 1, 4. Equilibrium tensile stress-strain Isotherms were obtained at 25 C on dumbbell shaped specimens according to procedures described elsewhere (1, 4). The data were well correlated by linear regression to the empirical Mooney-Rivlin (6 ) relationship. The tensile behavior of the networks formed In solution was measured both on networks with the solvent present and on networks from which the oligomeric PEMS had been extracted. 

Figure 2. Typical stress-strain isotherms for PDMS networks prepared by tetra-functionally end-linking very short and relatively long chains.

Figure 5. Stress-strain isotherms obtained for bimodal (600-11,300), PDMS networks containing 75.2 mol % short chains (20).

Figure 6. Effect of temperature on the stress-strain isotherms exhibited by a tetra-functional bimodal (220-18,500) PDMS network containing 75 mol % short...

Hydroxyl-terminated POET chains are end-linked into noncrystallizable trifunctional networks using an aromatic triisocyanate. The networks thus obtained are studied with regard to their stress-strain isotherms. The analysis of the temperature coefficient of PDET in terms of the RIS model confirms the results obtained from NMR studies, according to which the gauche states... 

Mark,J.E., Flory.P.J. Stress-strain isotherms for poly-(dimethylsiloxane) networks. J. Appl. Phys. 37,4635-4639 (1966). 

Figure 8.4 Stress-strain isotherms for PDMS networks reinforced with in situ generated titania particles.39 Each curve is labeled with the wt % of filler introduced, and filled circles locate results used to test for reversibility.

Figyre 3.16 Mooney-Rivlin plots [Eq. (3.38)] showing the effect of the temperature on stress-strain isotherms for model PDMS networks (15,16). The filled circles represent the reversibility of the elastic measurements, and the vertical lines locate the fracture points. (From Ref. 15.)... 

Uniaxial stress-strain measurements are often used to characterize polymer networks both in the dry state and in equilibrium with a diluent. The analysis of the stress-strain isotherms is usually performed in terms of the reduced force... 

TABLE 29.1. Parameters of the stress-strain isotherms caicuiated from the fit of the Fiory-Erman modei for different networks systems [ 114],... 

TABLE 29.4. Parameters of the stress-strain isotherms for PDMS modei networks caicuiated from the entanglement model (Eq. (29.46)) [54]. 

FIGURE 32.8. Stress-strain isotherms for PDMS networks filled with in situ generated silica for both elongation (a < 1) and compression (a >1). The filled points represent the data used to test for reversibility. From [69] 1991 American Chemical Society. 

Unusual properties are also obtained for networks prepared and studied in the opposite way, specifically cross linking in the dry state and then swelling the network prior to the measurements of mechanical properties. The approach to elastic equilibrium is more rapid and the stress-strain isotherms in elongation are closer to the form predicted by the... 

Many of the stress-strain isotherms of bimodal networks were obtained on PDMS elastomers in the vicinity of 25°C, a temperature sufficiently high to suppress strain-induced crystallization. These elongational isotherms show greatly improved the ultimate properties (figure 7.17). ° ... 

Schematic stress-strain isotherms in elongation for a unimodal elastomer in the Mooney-Rivlin representation of modulus against reciprocal elongation. The isotherms are represented as the dependence of the reduced stress ([f ] = f /(a - on reciprocal elongation. (f = f/A, f = elastic force, A = undeformed area, a = elongation). The top three are for a crystallizable network curve A for a relatively low temperature, B for an increased temperature, and C for the introduction of a swelling diluent. Isotherm D is for an unswollen unimodal network that is inherently noncrystallizable.

It is possible to characterize this non-Gaussian limited extensibility more quantitatively in a number of ways. The first involves the interpretation of limited chain extensibility in terms of the configurational characteristics of the PDMS chains making up the network structure. The upturn in modulus generally begins at approximately 60-70% of maximum chain extensibility. " This value is approximately twice that estimated previously- from stress-strain isotherms of elastomers that may have been undergoing strain-induced crystallization. 

More quantitative characterization of limited chain extensibility requires a non-Gaussian distribution function for the end-to-end separation, r, of the short network chains. The Fixman-Alben distribution was used to calculate stress-strain isotherms in elongation for bimodal PDMS networks. Good agreement was found between theory and experiment. Other non-Gaussian distribution functions have also been successfully used. The experimental isotherms can also be interpreted... 

Wen, J. Mark, J. E., Torsion Studies of Thermoelasticity and Stress-Strain Isotherms of Unimodal, Bimodal, and Filled Networks of Polyfdimethylsiloxane). Polym. J. 1994,26,151-157. 

Stress-strain isotherms have also been calculated with this approach. Examples are unimodal networks of polyethylene and POMS, " polymeric sulfur and seleniirm, short n-alkane chains, natural rubber, several polyoxides, and elastin, and bimodal networks of PDMS. It is possible to include excluded volume effects, in such simulations. In the case of the partially helical polymer polyoxymethylene, the simulations were used to resolve the overall distributions into contributions from imbroken rods, once-broken rods, twice-broken rods, and so on. It was also shown how applying stresses to the ends of chains of this typ>e can be used to bias the distributions in the direction of increased helical content and increased average end-to-end distances. In this sense, imposition of a stress has the same effect on the helix-coil equilibriirm as a decrease in temperature. ... 

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